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Tag 0AP5

36.55. Ind-quasi-affine morphisms

A bit of theory to be used later.

Definition 36.55.1. A scheme $X$ is ind-quasi-affine if every quasi-compact open of $X$ is quasi-affine. Similarly, a morphism of schemes $X \to Y$ is ind-quasi-affine if $f^{-1}(V)$ is ind-quasi-affine for each affine open $V$ in $Y$.

An example of an ind-quasi-affine scheme is an open of an affine scheme. If $X = \bigcup_{i \in I} U_i$ is a union of quasi-affine opens such that any two $U_i$ are contained in a third, then $X$ is ind-quasi-affine. An ind-quasi-affine scheme $X$ is separated because any two affine opens $U, V$ are contained in a separated open subscheme of $X$, namely $U \cup V$. Similarly an ind-quasi-affine morphism is separated.

Lemma 36.55.2. The property of being ind-quasi-affine is stable under base change.

Proof. Let $f : X \to Y$ be an ind-quasi-affine morphism. Let $Z$ be an affine scheme and let $Z \to Y$ be a morphism. To show: $Z \times_Y X$ is ind-quasi-affine. Let $W \subset Z \times_Y X$ be a quasi-compact open. We can find finitely many affine opens $V_1, \ldots, V_n$ of $Y$ and finitely many quasi-compact opens $U_i \subset f^{-1}(V_i)$ such that $Z$ maps into $\bigcup V_i$ and $W$ maps into $\bigcup U_i$. Then we may replace $Y$ by $\bigcup V_i$ and $X$ by $\bigcup W_i$. In this case $f^{-1}(V_i)$ is quasi-compact open (details omitted; use that $f$ is separated) and hence quasi-affine. Thus now $X \to Y$ is a quasi-affine morphism (Morphisms, Lemma 28.12.3) and the result follows from the fact that the base change of a quasi-affine morphism is quasi-affine (Morphisms, Lemma 28.12.5). $\square$

Lemma 36.55.3. The property of being ind-quasi-affine is fpqc local on the base.

Proof. Let $f : X \to Y$ be a morphism of schemes. Let $\{g_i : Y_i \to Y\}$ be an fpqc covering such that the base change $f_i : X_i \to Y_i$ is ind-quasi-affine for all $i$. We will show $f$ is ind-quasi-affine. Namely, let $U \subset X$ be a quasi-compact open mapping into an affine open $V \subset Y$. We have to show that $U$ is quasi-affine. Let $V_j \subset Y_{i_j}$, $j = 1, \ldots, m$ be affine opens such that $V = \bigcup g_{i_j}(V_j)$ (exist by definition of fpqc coverings). Then $V_i \times_Y X \to V_i$ is ind-quasi-affine as well. Hence we may replace $Y$ by $V$ and $\{g_i : Y_i \to Y\}$ by the finite covering $\{V_j \to V\}$. We may replace $X$ by $U$, because $V_j \times_Y U \subset V_j \times_Y X$ is open and hence $V_j \times_Y U \to V_j$ is ind-quasi-affine as well (ind-quasi-affineness is inherited by opens). Hence we may assume $X$ is quasi-compact and $Y$ affine. In this case we have to show that $X$ is quasi-affine and we know that $X_i$ is quasi-affine. Thus the result follows from Descent, Lemma 34.20.20. $\square$

Lemma 36.55.4. A separated locally quasi-finite morphism of schemes is ind-quasi-affine.

Proof. Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes. Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact open. We have to show $U$ is quasi-affine. Since $U \to V$ is a separated quasi-finite morphism of schemes, this follows from Zariski's Main Theorem. See Lemma 36.38.2. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 16464–16552 (see updates for more information).

    \section{Ind-quasi-affine morphisms}
    \label{section-ind-quasi-affine}
    
    \noindent
    A bit of theory to be used later.
    
    \begin{definition}
    \label{definition-ind-quasi-affine}
    A scheme $X$ is {\it ind-quasi-affine} if every quasi-compact open of
    $X$ is quasi-affine. Similarly, a morphism of schemes $X \to Y$
    is {\it ind-quasi-affine} if $f^{-1}(V)$ is ind-quasi-affine
    for each affine open $V$ in $Y$.
    \end{definition}
    
    \noindent
    An example of an ind-quasi-affine scheme is an open of an affine scheme.
    If $X = \bigcup_{i \in I} U_i$ is a union of quasi-affine opens such that
    any two $U_i$ are contained in a third, then $X$ is ind-quasi-affine.
    An ind-quasi-affine scheme $X$ is separated because any two affine
    opens $U, V$ are contained in a separated open subscheme of $X$, namely
    $U \cup V$. Similarly an ind-quasi-affine morphism is separated.
    
    \begin{lemma}
    \label{lemma-base-change-ind-quasi-affine}
    The property of being ind-quasi-affine is stable under base change.
    \end{lemma}
    
    \begin{proof}
    Let $f : X \to Y$ be an ind-quasi-affine morphism. Let $Z$ be an affine
    scheme and let $Z \to Y$ be a morphism. To show: $Z \times_Y X$ is
    ind-quasi-affine. Let $W \subset Z \times_Y X$ be a quasi-compact open.
    We can find finitely many affine opens $V_1, \ldots, V_n$ of $Y$
    and finitely many quasi-compact opens $U_i \subset f^{-1}(V_i)$
    such that $Z$ maps into $\bigcup V_i$ and $W$ maps into $\bigcup U_i$.
    Then we may replace $Y$ by $\bigcup V_i$ and $X$ by $\bigcup W_i$.
    In this case $f^{-1}(V_i)$ is quasi-compact open (details omitted; use
    that $f$ is separated) and hence quasi-affine. Thus now $X \to Y$
    is a quasi-affine morphism (Morphisms, Lemma
    \ref{morphisms-lemma-characterize-quasi-affine}) and the result
    follows from the fact that the base change of a quasi-affine morphism
    is quasi-affine (Morphisms, Lemma
    \ref{morphisms-lemma-base-change-quasi-affine}).
    \end{proof}
    
    \begin{lemma}
    \label{lemma-descending-property-ind-quasi-affine}
    The property of being ind-quasi-affine is fpqc local on the base.
    \end{lemma}
    
    \begin{proof}
    Let $f : X \to Y$ be a morphism of schemes. Let $\{g_i : Y_i \to Y\}$
    be an fpqc covering such that the base change $f_i : X_i \to Y_i$
    is ind-quasi-affine for all $i$. We will show $f$ is ind-quasi-affine.
    Namely, let $U \subset X$ be a quasi-compact open mapping into
    an affine open $V \subset Y$. We have to show that $U$ is quasi-affine.
    Let $V_j \subset Y_{i_j}$, $j = 1, \ldots, m$ be affine opens
    such that $V = \bigcup g_{i_j}(V_j)$ (exist by definition of
    fpqc coverings). Then $V_i \times_Y X \to V_i$ is ind-quasi-affine
    as well. Hence we may replace $Y$ by $V$ and $\{g_i : Y_i \to Y\}$
    by the finite covering $\{V_j \to V\}$. We may replace $X$ by
    $U$, because $V_j \times_Y U \subset V_j \times_Y X$ is open and
    hence $V_j \times_Y U \to V_j$ is ind-quasi-affine as well
    (ind-quasi-affineness is inherited by opens).
    Hence we may assume $X$ is quasi-compact and $Y$ affine.
    In this case we have to show that $X$ is quasi-affine and we
    know that $X_i$ is quasi-affine. Thus the result follows from
    Descent, Lemma \ref{descent-lemma-descending-property-quasi-affine}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-separated-ind-quasi-affine}
    A separated locally quasi-finite morphism of schemes is ind-quasi-affine.
    \end{lemma}
    
    \begin{proof}
    Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes.
    Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact
    open. We have to show $U$ is quasi-affine. Since $U \to V$ is a
    separated quasi-finite morphism of schemes, this follows from
    Zariski's Main Theorem. See
    Lemma \ref{lemma-quasi-finite-separated-quasi-affine}.
    \end{proof}

    Comments (2)

    Comment #1785 by Laurent Moret-Bailly on January 8, 2016 a 5:32 am UTC

    Examples after the definition: an open of a quasi-projective scheme is not ind-quasi-affine in general! On the other hand, (an open of) a disjoint sum of quasi-affine schemes is, or more generally (an open of) a scheme with a filtered open covering by quasiaffines. (Any interesting examples of the latter?)

    Typo in proof of 36.49.2: $\bigcup V_i$.

    Comment #1788 by Laurent Moret-Bailly on January 9, 2016 a 8:28 am UTC

    Examples after the definition: an open of a quasi-projective scheme (e.g. $\mathbb{P}^1$) is not ind-quasi-affine in general! One could mention disjoint sums of quasi-affine schemes (but this is not a "new" example since $\coprod_i\mathrm{Spec}\,(R_i)$ is open in $\mathrm{Spec}\,(\prod_i R_i)$). More generally (an open of) a scheme with a filtered open covering by quasiaffines is ind-quasi-affine. Any interesting examples of the latter?

    Typo in proof of 36.49.2: $\bigcup V_i$.

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